Bounds in cohomology
The author introduces a measure of complexity for affine algebras and their finitely generated modules. For some affine local \(K\)-algebra \(R\) and a finitely generated \(R\)-module \(M\) with the presentation \(R^s @>f>> R^t \to M \to 0\) the complexity of \(M\) is at most \(d\) provided \(t, s \leq d\) and each entry of the matrix of \(f\) is the quotient of two polynomials of degree at most \(d.\) Then the author studies various cohomological operations and numerical invariants, uniformly bounded with respect to these complexities. The Lefschetz principle, most succinctly stated, is the following isomorphism of fields \[ {\mathbb{C}} \simeq \prod_U {\mathbb{F}}_p^{\text{alg}}, \] where \({\mathbb{F}}_p^{alg}\) denotes the algebraic closure of the \(p\)-element field \({\mathbb{F}}_p\), \(p\) a prime number, and \(U\) the non-principal ultrafilter on the set of primes. This enables the transfer of properties between various characteristics. The first application concerns the Bass conjecture and the ``new intersection theorem''. Its validity for affine local rings over an algebraically closed field of characteristic zero can be derived from its validity in the case of positive characteristic. -- The second application concerns the Zariski-Lipman conjecture in positive characteristic. While the conjecture is not true in positive characteristic, as shown by \textit{J.~Lipman} [see Am. J. Math. 87, 874-898 (1965; Zbl 0146.17301)], the author proves the following result in positive characteristic: Let \(R\) denote the local ring of a point \(P\) on a hypersurface over an algebraically closed field \(K\) such that the module of \(K\)-invariant derivations on \(R\) is free. Then \(P\) is a non-singular point, provided the characteristic is larger than some bound depending on the degree of the hypersurface. Reviewers remark: The new intersection theorem as well as the Bass conjecture is true in general as was finally shown by \textit{P. Roberts} [see C. R. Acad. Sci., Paris, Sér. I 304, 177-180 (1987; Zbl 0609.13010)]. Unfortunately this is not noted in the paper under review.
scientific article; zbMATH DE number 1445778
scientific article